Understanding mirror modifier
When you mirror, the vertices you create will be created on the other side of the axis, with the object origin as center. If you choose the X axis and create some vertices on the +X side, they will be mirrored on the -X side, but if you create vertices on the -X side, they will be mirrored on the +X side as well. To avoid any trespassing you can enable the Clipping option, that way you'll make sure that if you move a vertex close to the axis plan, it won't trespass on the other side, but it won't delete the vertices that are already on the other side, like what's happening in your case.
As for your second question, the mirroring currently happens on the X axis while you've selected the Y axis, but actually the Mirror modifier takes the local orientation into account, not the global orientation. You must have rotated your object in Object mode, so its local orientation is not aligned with the global anymore. What you can do is apply the rotation of your object in Object mode (CtrlA &gt; Rotation), then choose X in the Mirror modifier.